A Hochschild-Kostant-Rosenberg theorem for cyclic homology. (English) Zbl 1387.19004
The main result of the paper under review is to relate the authors’ previous work on the \(\ell\)-functor [M. Bökstedt and I. Ottosen, Fundam. Math. 162, No. 3, 251–275 (1999; Zbl 0952.55006)] to negative, ordinary, and periodic cyclic homology [J.-L. Loday, Cyclic homology. 2nd ed. Berlin: Springer (1998; Zbl 0885.18007)]. Results are stated for a commutative ring \(A\) over the field \({\mathbb{Z}}/2\). Then \(\ell (A)\) is defined as the free, commutative, unital \({\mathbb{Z}}/2\)-algebra on generators \(\delta (a)\), \(\phi (a)\), \(q(a)\) for \(a \in A\), and a generator \(u\), subject to a list of twelve relations, stated in the paper. When \(A\) is unital, there is a natural algebra homomorphism
\[
\psi : \ell (A) \to HC^{-}_{*} (A),
\]
where \(HC^{-}_* (A)\) denotes the negative cyclic homology of \(A\). The authors call \(A\) supplemented if \(A\) is endowed with an augmentation \(A \to k\) so that the composition \(k \to A \to k\) is the identity. Proven is that if \(A\) is smooth (over \({\mathbb{Z}}/2\)), supplemented, and of finite type, then \(\psi\) is an algebra isomorphism. Now let \(\ell^+ (A)\) be the free \(\ell (A)\)-module on generators \(\gamma (a)\), \(a \in A\) and \(v^i\), \(i = 0, \;1, \;2, \ldots \, \), modulo a list of nine relations. Proven is that there is a natural \(\ell (A)\)-linear map
\[
\psi^+ : \ell^+ (A) \to HC_* (A),
\]
where \(HC_*(A)\) denotes the (usual) cyclic homology of \(A\). When \(A\) is smooth, \(\psi^+\) is an isomorphism. Finally, let \(\ell^{\mathrm{per}} (A)\) be the free, commutative, unital \({\mathbb{Z}}/2\)-algebra on generators \(\psi (a)\), \(q(a)\), \(a \in A\), and \(u\), \(u^{-1}\), subject to a list of seven relations. Then there is an algebra homomorphism
\[
\psi^{\mathrm{per}} : \ell^{\mathrm{per}} (A) \to HC^{\mathrm{per}}_* (A),
\]
where \(HC^{\mathrm{per}}(A)\) denotes periodic cyclic homology. If \(A\) is smooth, supplemented, and of finite type, then \(\psi^{\mathrm{per}}\) is an isomorphism. Additionally there are natural isomorphisms of functors from unital commutative \({\mathbb{Z}}/2\)-algebras of finite type to unital graded commutative \({\mathbb{Z}}/2\)-algebras
\[
\ell \simeq L_0 ( HC^{-}_* ), \;\;\;\ell^{\mathrm{per}} \simeq L_0 ( HC^{\mathrm{per}}_* )
\]
and a natural isomorphism to graded \({\mathbb{Z}}/2\)-vector spaces
\[
\ell^+ \simeq L_0 ( HC_* ),
\]
where \(L_0\) denotes the zeroth derived functor.
Reviewer: Jerry M. Lodder (Las Cruces)
MSC:
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
| 18G50 | Nonabelian homological algebra (category-theoretic aspects) |
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